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Inverse Trig Functions & Integration © G. Battaly 2015 1 September 28, 2015 Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 5.7 Inverse Trig Functions and Integration HW: 5.7 # 5 13, 21 31, 47 GOAL: 1. Distinguish methods of integrating rational expressions using past procedures. 2. Recognize rational expressions that integrate to inverse trig functions. 3. Perform integrations that involve inverse trig functions. 5.7 Inverse Trig Functions and Integration Calculus Home Page Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Which can be integrated? How? 1. 2. 3. 4.

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Page 1: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 1

September 28, 2015

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

HW:  5.7  # 5 ­ 13,  21 ­ 31, 47

GOAL:1.  Distinguish methods of integrating rational expressions using past procedures.2.  Recognize rational expressions that integrate to inverse trig functions.3.  Perform integrations that involve inverse trig functions.

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Which can be integrated?  How?

1. 2.

3. 4.

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 2: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 2

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Which can be integrated?  How?

1. 2.

3. 4.

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Integrating Rational Expressions:1.  Is the denominator a monomial?

   >  If yes, divide and integrate.   >  If no, proceed to #2.

2.  If the quotient is rewritten as a product, is the exponent (­1)?   >  If yes, think ln and look for du/u   >  If no, think un and look for un du

3.  If none of above work, think inverse trig functions and look for     du/ √a2 ­ u2       or      du/ ( a2 + u2 )      or    du/ u √u2 ­ a2     

Previous

New

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 3: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 3

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Can we integrate this using        ∫ un du        ? 

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

Can we integrate this using        ∫ un du        ? 

Does not work.  The result is not less complicated than original.  So there is no progress.

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 4: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 4

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Integrating Rational Expressions:1.  Is the denominator a monomial?

   >  If yes, divide and integrate.   >  If no, proceed to #2.

2.  If the quotient is rewritten as a product, is the exponent (­1)?   >  If yes, think ln and look for du/u   >  If no, think un and look for un du

3.  If none of above work, think inverse trig functions and look for     du/ √a2 ­ u2       or      du/ ( a2 + u2 )      or    du/ u √u2 ­ a2     

NO

NO

Not undu   ­­> no du

YES

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

using trig:

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 5: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 5

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Is there a shorter way?

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

What does this form suggest?

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

θ

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 6: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 6

September 28, 2015

5.7   Inverse Trig Functions and  Integration

What does this form suggest?

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

hyp sidesinθ =  2x              1

θ

∴Use arc sin (___)

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

using formula:

5.7   Inverse Trig Functions and  Integration

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 7: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 7

September 28, 2015

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

using formula:

5.7   Inverse Trig Functions and  Integration

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Integrating Rational Expressions:1.  Is the denominator a monomial?

   >  If yes, divide and integrate.   >  If no, proceed to #2.

2.  If the quotient is rewritten as a product, is the exponent (­1)?   >  If yes, think ln and look for du/u   >  If no, think un and look for un du

3.  If none of above work, think inverse trig functions and look for     du/ √a2 ­ u2       or      du/ ( a2 + u2 )      or    du/ u √u2 ­ a2     

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 8: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 8

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Integrating Rational Expressions:1.  Is the denominator a monomial?

   >  If yes, divide and integrate.   >  If no, proceed to #2.

2.  If the quotient is rewritten as a product, is the exponent (­1)?   >  If yes, think ln and look for du/u   >  If no, think un and look for un du

3.  If none of above work, think inverse trig functions and look for     du/ √a2 ­ u2       or      du/ ( a2 + u2 )      or    du/ u √u2 ­ a2     

NO

Not du/u   ­­> no du

YES

yes

θtanθ =  a             b

∴Use arc tan (___)

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 9: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 9

September 28, 2015

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 10: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 10

September 28, 2015

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 11: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 11

September 28, 2015

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration formulas

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 12: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 12

September 28, 2015

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

5.7   Inverse Trig Functions and  Integration formulas

u=arc cosxdu=   dx         √1­x2   x      u1/√2   π/4   0     π/2

5.7   Inverse Trig Functions and  Integration

Calculus Home PageClass Notes:  Prof. G. Battaly, Westchester Community College, NY

Homework Part 1  

formulas

http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
http://www.battaly.com/calc/wcccalc1.htm
http://www.battaly.com/calc/hw/ch5s7_2008.htm
Page 13: Inverse Trig Functions & Integration · Inverse Trig Functions & Integration © G. Battaly 2015 2 September 28, 2015 5.7 Inverse Trig Functions and Integration Calculus Home Page

Inverse Trig Functions & Integration

© G. Battaly 2015 13

September 28, 2015


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